The Trispectrum in the Effective Theory of Inflation with Galilean symmetry
We calculate the trispectrum of curvature perturbations for a model of inflation endowed with Galilean symmetry at the level of the fluctuations around an FRW background. Such a model has been shown to posses desirable properties such as unitarity (up to a certain scale) and non-renormalization of the leading operators, all of which point towards the reasonable assumption that a full theory whose fluctuations reproduce the one here might exist as well as be stable and predictive.
The cubic curvature fluctuations of this model produce quite distinct signatures at the level of the bispectrum. Our analysis shows how this holds true at higher order in perturbations. We provide a detailed study of the trispectrum shape-functions in different configurations and a comparison with existent literature. Most notably, predictions markedly differ from their counterpart in the so called equilateral trispectrum configuration.
The zoo of inflationary models characterized by somewhat distinctive predictions for higher order correlators is already quite populated; what makes this model more compelling resides in the above mentioned stability properties.
- 1 Introduction
- 2 The Action to fourth order in perturbations
- 3 Trispectrum of curvature fluctuations
- 4 Conclusions
The inflationary paradigm  stands at the core of modern cosmology. It set out to solve several cosmological puzzles which pre-dated it and delivered further detailed predictions verified to be in close agreement with observations [2, 3, 4, 5]. The simplest models of single-field slow-roll inflation are, as of today, in agreement with data. On the other hand, it is natural to ask whether it is possible to find or embed an inflationary mechanism within UV complete theories such as string theory. One such a realization (DBI inflation) was put forward in [6, 7, 8], and has since been studied in great details (for other realizations see , [10, 11, 12, 13] and [14, 15, 16] for some reviews). However, the regime in which the DBI model can be trusted seems perilously close to what data might soon exclude.
Stopping a bit short of this last, quite ambitious, requirement, one could also aim at a model of inflation which is unitary, stable under quantum corrections and predictive. In  the authors propose
an inflationary set up in this direction, where the requirement of the so called Galileon symmetry (, see  ) grants all of the above properties:
- in flat space the (where the index counts space-time dimensions) Galilean terms are guaranteed to have second order equations of motion 
- the non-renormalization theorem for Galileon models  warrants that (at least in flate space) the coefficients which one writes down at first will not be largely modified by renormalization.
Crucially, these desirable properties are shown to approximately (up to corrections, where is the scale of the underlying theory) hold for the model in  also in the more realistic scenario where one deals with curved spacetime and couples to gravity. There exists an interesting energy regime where the Galileon interactions are important and the model is quite different from canonical slow-roll inflation. In this same regime both the metric fluctuations and terms with non-minimal coupling to gravity can be neglected and calculating higher order curvature correlators is greatly simplified.
The bispectrum of this model has been explored in : the non-linearity parameter generated can be detectable but the shape-functions peak in the equilateral configuration and thus one is not able to remove the large degeneracy with most other inflationary models at the level of the bispectrum. This, of course, leaves the appeal of the model just described untouched. One should then go look for another quantity at (possibly) observational reach, the trispectrum being the natural choice. This is precisely what we do in a paper  which is complementary to the work presented here.
On the other hand, if one is prepared to give away a little bit in terms of the cherished properties described above (this statement will be made more quantitative below), then there exist a model  which already at the bispectrum level generates interesting and quite distinct shape signatures. Combining the analysis of  with the study of the four point function, the trispectrum, one can really hope to fully characterize the predictions of this generalized Galileon model and disentangle it from a host of inflationary models with different, somewhat more canonical, signatures.
- has a smaller scale up to which the effective theory can be trusted
- with some assumptions (reasonable in light of  and other works), preserves the non-renormalization properties of the Galileon terms
- has a richer and more distinct set of predictions for non-Gaussianities
What has been proven concerning the background (and therefore the full theory) in  is more or less implicitly assumed at the onset of  because the latter is an effective field theory of fluctuations around a background rather than a effective field theory as a whole. The focus is on fluctuations and their properties.
Following  then, we study the four-point function of curvature perturbations in detail, we calculate the corresponding amplitude and plot the shape-functions of each interaction term in different configurations so as to ease the comparison with the existent literature on the subject and, at least at the level of shape-functions, remove as much degeneracy as possible.
On the other hand, the model we study is not quite as generic as it could have been in the spirit of : we are now dealing with a theory which is unitary (up to some scale) and has been further endowed with a symmetry which selects interaction operators and protects them from quantum corrections. This subset of theories is more stable and more predictive in that the symmetry shields us from large renormalization of the coefficients of interaction terms and keeps the number of leading interaction operators to a finite, small number.
The paper is organized as follows. In Sec. 2 we introduce the action for the model and expand it up to fourth order in fluctuations, we also perform a field redefinition which returns the action in a form that makes it suitable for employment within the Hamiltonian formulation of the Schwinger-Keldysh formalism. In Sec. 3, we compute the trispectrum of curvature fluctuations, study its amplitude and shape functions so as to draw a comparison with some of the results already present in the literature for other inflationary models, especially DBI inflation (for studies on primordial trispectrum in DBI, and more general models, and other inflation models, see, e.g., [39, 40, 41, 42, 43, 44, 46, 45, 47, 48, 49]). In Sec. 4 we conclude, and offer some further comments. In Appendix A we present the details of our field redefinition while in Appendix B we review the main steps that characterize the IN-IN formalism computation of the trispectrum and we list the final analytical results for the shape functions. In the following greek indices run from to , latin indices are spatial only, running from to .
2 The Action to fourth order in perturbations
2.1 Building Blocks of the Action
We are interested in a Lagrangian for scalar perturbations around an FRW background whose fluctuations satisfy the Galilean symmetry. The model can be written down by listing all possible Galilean invariant terms which non-linearly realize Lorentz symmetry.
Alternatively, one can write down Lorentz invariant contributions in terms of where linearly realizes the symmetry . We shall proceed according to the latter prescription. What we are after is the number of independent operators generated by products of traces of the form:
where each is to be understood as a trace of whatever product is inside the brackets, , and indices are raised with the metric . It can be shown that, once all the terms of the form with are known, all the rest in Eq. (1) can be expressed as linear combinations of products of these building blocks .
With fluctuations in mind, one removes the background value from each trace below:
where the expressions of the type are meant as shortcuts for (greek indices run from to , latin indices are spatial only) . It is easy to check that Eq. (LABEL:an) represents a basis which spans the entire space (up to order ) of the operators for any .
In what follows we do not consider contributions coming from a single trace e.g. ; the reason is that these will account  for the so-called minimal covariant Galileon terms [22, 23, 24] and those, their expression being known, can be put in at a later stage. More explicitly, the single trace operators can be written as sums of multiple trace operators and minimal Galileon terms. The former are already all accounted for in Eq. (LABEL:an) while the latter can be shown to be subdominant whenever generic operators are switched on, as is the case here .
With Eq. (LABEL:an) handy, one sets about to write down all the leading independent interactions at third and fourth order in perturbations. Particular care must be exerted so as to have that, at each perturbative order, the coefficient multiplying a given interaction term does not appear already at lower perturbative order. Such an occurrence is not problematic by itself but, as is known , the third and fourth order contribution of a piece like would be suppressed by a scale larger than the one that suppresses the quadratic fluctuations. The resulting three and four-point functions would therefore be negligible with respect to the contribution coming from interactions suppressed by .
Indeed, what is natural to assume at this stage is that:
Every independent interaction term at each given perturbative order is suppressed by a common scale . This is possible if each coefficient which corresponds to each independent interaction appears at given perturbative order and nowhere else 111 That is, appears as multiplying a leading operator..
This requirement is relatively easy to implement. Using the building blocks in Eq. (LABEL:an), what one wants is independent combinations of product of multiple trace operators whose first contribution starts at a given perturbative order
where the ’s are constant (to first approximation in generalized slow-roll) coefficients. As an example, consider:
The coefficients are all different and appear only once in our total Lagrangian. Applying this algorithm to the fullest, one obtains the interactions Lagrangians below.
2.2 Interaction Lagrangian
We write below the expression for the action in :
The second order Lagrangian is
If one were to write down the equations of motion for, say, above, those would surely include terms like third time derivatives of the scalar ; that is because the action above comprises higher time derivative terms. If taken literally, this fact leads in turn to the excitation of unstable modes in the dynamics and renders the Hamiltonian of the system unstable . In perturbation theory, it is however possible to treat this instability by essentially connecting the “higher derivative expansion” with the perturbative expansion in the field . In the case at hand the latter procedure corresponds to performing an order-by-order (in perturbations) field redefinition. If one stops at third order in fluctuations, it can further be shown that a field redefinition is equivalent (up to corrections) to the repeated use of the -derived equation of motion to get rid of the time derivatives in excess. This is where we part ways with the analysis in : there the bispectrum was the target and the use of the e.o.m.’s was entirely justified to this aim, here we also want to calculate the trispectrum so we have to undergo (as also recognized in ) a full fledged field redefinition: the corrections do matter for our purposes.
Let us redefine the field as follows
where , . We are computing the action up to fourth order in the field fluctuations, therefore we do not need to go any further than . If we plug the expansion in Eq. (9) in the action up to fourth order in , we have
where, on the right-hand side, the lower index on indicates the perturbative order whereas the upper index reminds us from which order in that specific contribution arises. The “…” indicates contributions that are beyond fourth order and which we do not need for our computation of the trispectrum. The third and fourth order action in the newly redefined field are then
We will now determine and in such a way that the final equations of motion will be at most second in time derivatives and, in the process, derive the final expressions for the third and fourth order Lagrangian.
2.2.1 Cubic Lagrangian
Upon expanding Eq. (2.2), one can verify that the terms in that are expected to give rise to unwanted higher order time derivatives in the equation of motion for are
It is straightforward to check that the following field redefinition grants a cancellation of the above terms thanks to some of the contributions from
The final expression for the action to third order is obtained by summing the leftover terms from the two contributions in Eq.(14) (while the fourth order Lagrangian is generally affected by , no contributions to the cubic Lagrangian can possibly arise from )
We can now move on to the quartic Lagrangian.
2.2.2 Quartic Lagrangian
The fourth order action is computed from Eq. (15) with as given in Eq. (2.2.1). The expression for is to be determined in such a way that some of the contributions from will cancel the higher order time derivative terms from , and .
The process of computing is straightforward but rather long, therefore we report the details in Appendix A. Before providing the final result for the quartic Lagrangian, it is useful to make some considerations about the mass scales characterizing the action. Let us consider our initial Lagrangian (2.2)-(2.2), with coefficients and , respectively for third and fourth order contributions. The field redefinition (in the form of ) introduces extra interactions at the quartic level that are proportional to . Therefore, our Lagrangian after field redefinition has the form
Introducing the canonically normalized field , we have
We restate the assumption (as in ) that all operators multiplied by a coefficient that appears at a given order for the first time, are suppressed by a common scale . Eq. (24) can then be rewritten as
where we defined the scales
At this stage one can quickly relate the expression for to the one for
where if we are to make sense of perturbation theory. This is equivalent to stating that the scales of suppression are related by . In other words, at each perturbative order, all operators multiplied by coefficients which appear for the first time at that order, are leading compared to operators that are multiplied by coefficients that appear for the first time at a lower order. Based on these considerations, we can safely neglect all operators proportional to in the final fourth order Lagrangian which then reads:
where are linear combinations of the initial coefficients (we report their exact definitions in Appendix A). The final expression for the quartic action is now free of higher order time derivative terms and suitable for utilizing the Schwinger-Keldysh formalism [36, 37, 38] in the standard Hamiltonian formulation (as in ) for the computation of the trispectrum of curvature fluctuations. Eq. (2.2.2) is one of our main results.
We want to stress here that the field redefinition has not rendered the theory ghost-free, our procedure simply organizes the perturbative expansion in such a way as to have perturbations (up to a given order) to be consistent up to the scale . This is, after all, just an effective theory; if one wants to take the theory beyond this point, that is, above , one should first provide arguments as to why the ghost is not there also in that regime.
2.2.3 Interactions Hamiltonian
At third order in the field fluctuations, the interaction Hamiltonian is given by . This relation does not generally hold true at fourth order. We quickly review the procedure for deriving (see e.g. , whose notation we borrow in this section). Consider a Lagrangian of the type:
where are function of and its spatial derivatives (the subscript indicates whether a given function is order zero () linear () and so on in ). Introducing the conjugate momentum
we can derive the perturbative expansion, up to third order, of in terms of powers of . The total Hamiltonian reads:
which is split into quadratic and interactions parts
In the Schwinger-Keldysh formalism, is a function of and as in the interaction picture ( and ). Replacing
in the interaction Hamiltonian, one can derive the following relations
In our specific case, the corrections to in (35) are proportional to . From our previous discussion on mass hierarchies, which lead to the relation , we conclude that these corrections are subleading compared the terms in (regulated by ). In what follows we will therefore set for the quartic order interaction Hamiltonian.
3 Trispectrum of curvature fluctuations
We now have all the ingredients necessary in order to compute the trispectrum of curvature fluctuations
where we Fourier expanded the curvature fluctuation :
Whenever metric fluctuations are neglected, as is the case here, the curvature is linearly related to our field fluctuations :
In particular, at leading order approximation in slow-roll, one can then essentially read off the interactions from the Lagrangian. Upon quantization, the curvature is expressed in terms of creation and annihilation operators
Using conformal time variable , it is easy to derive from the quadratic fluctuations that the eigenfunctions are
where Bunch-Davies vacuum has been assumed. The power spectrum of reads
The diagrammatic computation in the Schwinger-Keldysh formalism is outlined in Appendix B. At tree-level, is given by the sum of two distinct diagrams: a so-called scalar-exchange diagram, from , and a contact-interaction diagram from (see Fig. 1). It turns out that the former contributions are subdominant compared to the latter. This can be easily shown by comparing the trispectrum amplitude from the two diagrams.
The amplitude of the trispectrum is conventionally described by a parameter, , defined as the result of the normalization of the trispectrum to the third power of the power spectrum. has a specific momentum dependence (related to the particular form of the interaction Hamiltonian) that we can only find by computing the interaction diagrams.
However, knowing the expressions of the eigenfunctions and of the interaction Hamiltonian, dimensional analysis quickly leads us to the parametric dependence of , which is necessary in order to estimate how big a contribution to the trispectrum a given diagram will provide. If we identify the contributions to the amplitude from the contact-interaction and from the scalar-exchange diagrams respectively as and , we have
Let us compare and :
which we have shown to be true (Eq. (27)). The contribution to the total trispectrum coming from terms represented by the contact-interaction diagram is then much larger than the contribution from the scalar-exchange side. We will therefore focus our attention on the former type of terms in the rest of the paper.
3.1 Shapes and Amplitudes from contact-interaction diagrams
Let us rewrite the Lagrangian in Eq. (2.2.2) in a schematic form
where we indicated operators by (e.g. and so on). We calculated the contributions from all of the interactions . Their analytic expressions are provided in Appendix B in the form of the shape functions (see Eqs. (115) through (122)). The the total trispectrum then reads:
The ’s depend on the external momenta which, because of momentum conservation, are oriented so as to form a closed tetrahedron. The total number of independent variables is six. We select them to be , , , , and , where and . One of the may be factored out in an overall normalization factor (we are in a scale invariant case), leaving us with five variables. This is of course too large a number for a plot of the momentum dependence of the .
One possibility is to select several different configurations in which the number of variables is narrowed down to two. We consider the configurations defined for the first time in  and applied to models therein (and later often times employed by other authors, see e.g. [29, 53]), so as to provide a clear comparison between results for the trispectrum in our model and the corresponding outcome in other inflationary models.
A first glance at our results, from the qualitative appearance of the plots of the for each configuration, one realizes the shape functions can be classified in two main categories. The analytical significance of these two categories is related to whether the respective vertices include or not the operator . It is also important to observe that the coefficients through in Eq. (2.2.2) are not all independent from one another. In fact, as one can easily check by looking at their definitions in terms of the original coefficients , only four out of the nine are independent. One therefore expects no more than four independent contributions to . We select a basis of independent coefficients and collect the operators accordingly.
The coefficients are linearly independent from one another, therefore they represent a natural candidate basis for the ensemble . In this basis, the Lagrangian becomes
From the analytic form of the diagrams with vertices and (Eq. (121)), we can immediately notice that gives a null contribution to the trispectrum. This means that we are left with three independent coefficients instead of four, which will generate three distinct contributions to , along with the corresponding shape functions. These shape functions are plotted in Figs. 2-3.
The expression of the total trispectrum is now:
The form of each of the three contributions is
where the ’s are functions of momentum which can be read off from Eqs.(115) through (122), using Eqs.(47)-(49). We consider the four momentum configurations introduced in . It turns out that, in all configurations, and are qualitatively very similar (although their analytic expressions do not coincide), whereas differs from them. As anticipated before Eq. (46), these similarities/differences are related to the operator : the vertex producing the contribution, unlike the ones producing and , does not contain . We will comment on the plots of and , in Figs. 2 and 3 respectively (all the features of are shared by ).
We provide below a configuration-by-configuration analysis of the results.
The four external momenta have equal length (). The leftover variables are and , which we normalize to . Triangular inequalities restrict their domains. If we define and , then and . As we can see from Fig. 1, is constant in this configuration, a functional behaviour that is also typical of contact-interaction diagrams in models .
On the other hand, the shape of (Fig. 2) is not a plateau and it very much resembles a shape that arises from contact interaction diagrams with the operator in Ghost Inflation ; a similar shape was found within the Effective Field Theory (EFT) approach to single-field inflation in , both from scalar-exchange (arising specifically from interactions) and from contact-interaction contributions (e.g. from and ).
We stress here that the aim of the paper is manifold. One is certainly after distinct features in higher order correlators, and indeed, we do find here signatures which are not present in quite general models such as models. On the other hand, we complement these findings with the fact that they occur within a theory which is, from the effective quantum field theory point of view, stable and predictive. In some respects this is in contradistinction to the approach of generic models (DBI being clearly an exception) and to the effective theory of inflation in its most comprehensive disguise . It is the combined (signatures & stability)-oriented approach which is at the heart of our work here.
The tetrahedron has , and . The variables are and with domains and respectively (also ). In this configuration, produces a shape which is very similar to the ones we can observe from contact interactions in models, whereas is a slightly different version of it in such that, unlike , it is not constant along the axes. also shares some features with the shapes from scalar-exchange diagrams in models (also characterized by a non constant values for ) however, unlike the latter, it is not null in or .
Specialized planar configuration.
In this limit and can be expressed in terms of and (both defined in the interval )
We considered, without loss of generality, the positive sign solution for our plots. Notice that, again, is very close to the contact interaction shapes for models, whereas is quite similar to the shape functions arising in these models from scalar-exchange diagrams.
Planar limit double-squeezed configuration.
In this configuration and the tetrahedron is going to be flattened. This allows to write in terms of and